TITLE: CW structures on unitary sphere of a banach/Hilbert space
QUESTION [5 upvotes]: Does the  unit sphere in a Banach  space carries a structure of  a CW complex? What about Finsler Manifolds?

REPLY [12 votes]: The answer is no in infinite dimension (and yes in the finite dimensional case, of course). Because the sphere in a Banach space is Baire. But an infinite dimensional CW-complex is not, being a countable union of finite dimensional skeletons.